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Hyperinterpolation on the sphere
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posted on 2023-06-07, 21:26 authored by Kerstin Hesse, Ian H SloanIn this paper we survey hyperinterpolation on the sphere $\\mathbb{S}^d$, $d\\geq 2$. The hyperinterpolation operator $L_n$ is a linear projection onto the space $\\mathbb{P}_n(\\mathbb{S}^d)$ of spherical polynomials of degree $\\leq n$, which is obtained from $L_2(\\mathbb{S}^d)$-orthogonal projection onto $\\mathbb{P}_n(\\mathbb{S}^d)$ by discretizing the integrals in the $L_2(\\mathbb{S}^d)$ inner products by a positive-weight numerical integration rule of polynomial degree of exactness $2n$. Thus hyperinterpolation is a kind of `discretized orthogonal projection' onto $\\mathbb{P}_n(\\mathbb{S}^d)$, which is relatively easy and inexpensive to compute. In contrast, the $L_2(\\mathbb{S}^d)$-orthogonal projection onto $\\mathbb{P}_n(\\mathbb{S}^d)$ cannot generally be computed without some discretization of the integrals in the inner products; hyperinterpolation is a realization of such a discretization. We compare hyperinterpolation with $L_2(\\mathbb{S}^d)$-orthogonal projection onto $\\mathbb{P}_n(\\mathbb{S}^d)$ and with polynomial interpolation onto $\\mathbb{P}_n(\\mathbb{S}^d)$: we discuss the properties, estimates of the operator norms in terms of $n$, and estimates of the approximation error. We also present a new estimate of the approximation error of hyperinterpolation in the Sobolev space setting, that is, $L_n:H^t(\\mathbb{S}^d)\\rightarrow H^s(\\mathbb{S}^d)$, with $t\\geq s\\geq 0$ and $t>d/2$, where $H^s(\\mathbb{S}^d)$ is for integer $s$ roughly the Sobolev space of those functions whose generalized derivatives up to the order $s$ are square-integrable.
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Publication status
- Published
Publisher
Chapman & Hall/CRCPage range
213-248Book title
Frontiers in Interpolation and Approximation (Dedicated to the memory of Ambikeshwar Sharma)ISBN
978-1-584-88636-5Department affiliated with
- Mathematics Publications
Full text available
- No
Peer reviewed?
- Yes
Editors
J Szabados, Ram N Mohapatra, N K Govil, Zuhair Nashed, H N MhaskarLegacy Posted Date
2012-02-06Usage metrics
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